SOLUTION: factor completely. if a polynomial is prime , state this. 10a^2-11a+3

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Question 883247: factor completely. if a polynomial is prime , state this.
10a^2-11a+3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 10a%5E2-11a%2B3, we can see that the first coefficient is 10, the second coefficient is -11, and the last term is 3.


Now multiply the first coefficient 10 by the last term 3 to get %2810%29%283%29=30.


Now the question is: what two whole numbers multiply to 30 (the previous product) and add to the second coefficient -11?


To find these two numbers, we need to list all of the factors of 30 (the previous product).


Factors of 30:
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to 30.
1*30 = 30
2*15 = 30
3*10 = 30
5*6 = 30
(-1)*(-30) = 30
(-2)*(-15) = 30
(-3)*(-10) = 30
(-5)*(-6) = 30

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -11:


First NumberSecond NumberSum
1301+30=31
2152+15=17
3103+10=13
565+6=11
-1-30-1+(-30)=-31
-2-15-2+(-15)=-17
-3-10-3+(-10)=-13
-5-6-5+(-6)=-11



From the table, we can see that the two numbers -5 and -6 add to -11 (the middle coefficient).


So the two numbers -5 and -6 both multiply to 30 and add to -11


Now replace the middle term -11a with -5a-6a. Remember, -5 and -6 add to -11. So this shows us that -5a-6a=-11a.


10a%5E2%2Bhighlight%28-5a-6a%29%2B3 Replace the second term -11a with -5a-6a.


%2810a%5E2-5a%29%2B%28-6a%2B3%29 Group the terms into two pairs.


5a%282a-1%29%2B%28-6a%2B3%29 Factor out the GCF 5a from the first group.


5a%282a-1%29-3%282a-1%29 Factor out 3 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%285a-3%29%282a-1%29 Combine like terms. Or factor out the common term 2a-1


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Answer:


So 10a%5E2-11a%2B3 factors to %285a-3%29%282a-1%29.


In other words, 10a%5E2-11a%2B3=%285a-3%29%282a-1%29.


Note: you can check the answer by expanding %285a-3%29%282a-1%29 to get 10a%5E2-11a%2B3 or by graphing the original expression and the answer (the two graphs should be identical).